Formula for the Area of a Circle Segment
Question
The distance from the center of the circle to the edge is defined as the
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Formula for the Area of a Circle Segment

Hi, I am Eric Stone from South Burlington High School in South Burlington, Vermont, here for About.com and today, we are going to find the area of a sector of a circle.
What is the Sector of a Circle?

Well first things first. In order to find the area of a sector of a circle, we have to define a few terms. First, what the heck is a sector of a circle? Well, it is simply what most people think of as a piece of the pie -- a piece of pizza. You have a cut that goes from the center out to the edge in both directions, and it is important that it does come from the center of the circle. If it does not, it is not a sector of a circle.

The distance from the center to edge is defined as the radius. You will probably notice that the area of a sector of a circle depends on how big of an angle you cut the sector in. That angle is called theta.
Circle Sector Area Formula

Put it all together and to find the area of a sector of a circle, we only need to know one formula: the area equals one half r squared theta. R being the distance from the center of the circle out to the edge and theta being the angle -- but this is really important -- that angle has to be expressed in radians. Not degrees.
Convert Degrees to Radians

So how do we handle that? Well, imagine I have an angle that is, say, 45 degrees, and I want to convert that over to radians. The only thing you have to remember is that 180 degrees equals pi radians. Once you remember that, converting from degrees to radians is fairly straightforward. You take the degrees and multiply the fact that pi radians is 180 degrees. Our 45 degree angle simplifies to one fourth pi radians. It is important to remember that usually, your answer in radians is going to be in terms of pi. That is a key.
Circle Sector Area in Perspective

So imagine you are at a pizza party and you want to make sure that two kids get the same amount of pizza even if the pizza sizes are not the same. Well, here we have two pizzas -- one which is 12 inches in radius and the other which is 9 inches in radius. We are going to use 45 degrees as the angle of the cut of the slice and we want to know what angle do I need to cut the smaller pizza at so that both kids get the same area of pizza.

Well, the first thing I need to do is I need to make sure that I get my 45 degrees in radians. And we have already done that -- 45 degrees times pi radians over 180 degrees gives me one fourth pi radians. So I can substitute that right in for my theta. And my radius was defined to be 12 inches so I will put that right there. One half times 12 inches squared times one fourth pi gives me 56.5 square inches of pizza.

We want to make sure that our other guest gets 56.5 square inches of pizza. Well, this pizza is only 9 inches in radius. No problem. The area we want to give them is 56.5 square inches and we know that that will be equal to one half r squared theta. So, 56.5 square inches will be equal to one half times nine squared times the angle theta that I want to cut it at. If I solve for theta by first squaring 9 and dividing by two, I'll have 56.5 square inches equals 40.5 times theta.

Dividing both sides by 40.5 gives me 1.4 radians is the angle -- not much use to the average consumer! So what we are going to do is convert those radians right back to degrees, which most people are familiar with. We'll take the 1.4 radians and multiply by the fact that 180 degrees is pi radians and, lo and behold, I know that I need to cut that pizza at 80 degrees!

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